DDR_klplate

Implementation of the DDR scheme for the Kirchoff-Love plate problem. More...

Classes
struct	HArDCore2D::KirchhoffLove
	Assemble a Kirchhoff-Love plate problem. More...

Typedefs
typedef Eigen::SparseMatrix< double >	HArDCore2D::KirchhoffLove::SystemMatrixType
typedef std::function< double(const Eigen::Vector2d &)>	HArDCore2D::KirchhoffLove::ForcingTermType
typedef std::function< double(const Eigen::Vector2d &)>	HArDCore2D::KirchhoffLove::DeflectionType
typedef std::function< Eigen::Vector2d(const Eigen::Vector2d &)>	HArDCore2D::KirchhoffLove::GradientDeflectionType
typedef std::function< Eigen::Matrix2d(const Eigen::Vector2d &)>	HArDCore2D::KirchhoffLove::MomentTensorType
typedef std::function< double(const Eigen::Vector2d &, const Edge &)>	HArDCore2D::KirchhoffLove::MomentTensorEdgeDerivativeType
typedef XDivDiv::ConstitutiveLawType	HArDCore2D::KirchhoffLove::ConstitutiveLawType

Functions
	HArDCore2D::KirchhoffLove::KirchhoffLove (const PlatesCore &platescore, const ConstitutiveLawType &law, bool use_threads, std::ostream &output=std::cout)
	Constructor.
size_t	HArDCore2D::KirchhoffLove::dimensionSpace () const
	Returns the dimension of the moment + deflection space.
size_t	HArDCore2D::KirchhoffLove::nbSCDOFs () const
	Returns the number of statically condensed DOFs (here, the cell moments DOFs)
size_t	HArDCore2D::KirchhoffLove::sizeSystem () const
	Returns the size of the statically condensed system.
const XDivDiv &	HArDCore2D::KirchhoffLove::xDivDiv () const
	Returns the space XDivDiv.
const GlobalDOFSpace	HArDCore2D::KirchhoffLove::polykm2Th () const
	Returns the space \(\mathbb{P}^{k-2}(\mathcal{T}_h)\).
const SystemMatrixType &	HArDCore2D::KirchhoffLove::systemMatrix () const
	Returns the linear system matrix.
SystemMatrixType &	HArDCore2D::KirchhoffLove::systemMatrix ()
	Returns the linear system matrix.
const Eigen::VectorXd &	HArDCore2D::KirchhoffLove::systemVector () const
	Returns the linear system right-hand side vector.
Eigen::VectorXd &	HArDCore2D::KirchhoffLove::systemVector ()
	Returns the linear system right-hand side vector.
const SystemMatrixType &	HArDCore2D::KirchhoffLove::scMatrix () const
	Returns the static condensation recovery operator.
Eigen::VectorXd &	HArDCore2D::KirchhoffLove::scVector ()
	Returns the static condensation rhs.
const double &	HArDCore2D::KirchhoffLove::stabilizationParameter () const
	Returns the stabilization parameter.
double &	HArDCore2D::KirchhoffLove::stabilizationParameter ()
	Returns the stabilization parameter.
void	HArDCore2D::KirchhoffLove::assembleLinearSystem (const ForcingTermType &f, const DeflectionType &u, const GradientDeflectionType &grad_u)
	Assemble the global system.
Eigen::VectorXd	HArDCore2D::KirchhoffLove::interpolateDeflection (const DeflectionType &u, int deg_quad=-1) const
	Interpolate deflection.
double	HArDCore2D::KirchhoffLove::computeNorm (const Eigen::VectorXd &v) const
	Compute the discrete norm.

Variables
static const double	HArDCore2D::PI = boost::math::constants::pi<double>()
static KirchhoffLove::DeflectionType	HArDCore2D::trigonometric_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::trigonometric_grad_u
static KirchhoffLove::MomentTensorType	HArDCore2D::trigonometric_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::trigonometric_dx_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::trigonometric_dy_hess_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::trigonometric_grad_Delta_u
static KirchhoffLove::MomentTensorEdgeDerivativeType	HArDCore2D::trigonometric_hess_u_DE
static KirchhoffLove::ForcingTermType	HArDCore2D::trigonometric_divdiv_hess_u
static KirchhoffLove::DeflectionType	HArDCore2D::quartic_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::quartic_grad_u
static KirchhoffLove::MomentTensorType	HArDCore2D::quartic_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::quartic_dx_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::quartic_dy_hess_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::quartic_grad_Delta_u
static KirchhoffLove::MomentTensorEdgeDerivativeType	HArDCore2D::quartic_hess_u_DE
static KirchhoffLove::ForcingTermType	HArDCore2D::quartic_divdiv_hess_u
static KirchhoffLove::DeflectionType	HArDCore2D::biquartic_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::biquartic_grad_u
static KirchhoffLove::MomentTensorType	HArDCore2D::biquartic_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::biquartic_dx_hess_u
static KirchhoffLove::MomentTensorType	HArDCore2D::biquartic_dy_hess_u
static KirchhoffLove::GradientDeflectionType	HArDCore2D::biquartic_grad_Delta_u
static KirchhoffLove::MomentTensorEdgeDerivativeType	HArDCore2D::biquartic_hess_u_DE
static KirchhoffLove::ForcingTermType	HArDCore2D::biquartic_divdiv_hess_u

Detailed Description

Implementation of the DDR scheme for the Kirchoff-Love plate problem.

Typedef Documentation

ConstitutiveLawType

typedef XDivDiv::ConstitutiveLawType HArDCore2D::KirchhoffLove::ConstitutiveLawType

DeflectionType

typedef std::function<double(const Eigen::Vector2d &)> HArDCore2D::KirchhoffLove::DeflectionType

ForcingTermType

typedef std::function<double(const Eigen::Vector2d &)> HArDCore2D::KirchhoffLove::ForcingTermType

GradientDeflectionType

typedef std::function<Eigen::Vector2d(const Eigen::Vector2d &)> HArDCore2D::KirchhoffLove::GradientDeflectionType

MomentTensorEdgeDerivativeType

typedef std::function<double(const Eigen::Vector2d &, const Edge &)> HArDCore2D::KirchhoffLove::MomentTensorEdgeDerivativeType

MomentTensorType

typedef std::function<Eigen::Matrix2d(const Eigen::Vector2d &)> HArDCore2D::KirchhoffLove::MomentTensorType

SystemMatrixType

typedef Eigen::SparseMatrix<double> HArDCore2D::KirchhoffLove::SystemMatrixType

Function Documentation

assembleLinearSystem()

void KirchhoffLove::assembleLinearSystem	(	const ForcingTermType &	f,
		const DeflectionType &	u,
		const GradientDeflectionType &	grad_u
	)

Assemble the global system.

Parameters

f	Forcing term
u	Deflection (for BC)
grad_u	Gradient of deflection (for BC)

computeNorm()

double KirchhoffLove::computeNorm

(

const Eigen::VectorXd &

v

)

const

Compute the discrete norm.

Parameters

v	The vector

dimensionSpace()

size_t HArDCore2D::KirchhoffLove::dimensionSpace ( ) const

inline

Returns the dimension of the moment + deflection space.

interpolateDeflection()

Eigen::VectorXd KirchhoffLove::interpolateDeflection	(	const DeflectionType &	u,
		int	deg_quad = -1
	)		const

Interpolate deflection.

Parameters

u	The function to interpolate
deg_quad	The degree of the quadrature rules (equal to \(2(k+1)\) with \(k\) equal to the degree of the complex by default)

KirchhoffLove()

KirchhoffLove::KirchhoffLove	(	const PlatesCore &	platescore,
		const ConstitutiveLawType &	law,
		bool	use_threads,
		std::ostream &	output = std::cout
	)

Constructor.

Parameters

platescore	Core for the DDR space sequence
law	Constitutive law: law(sigma)+hess u = 0
use_threads	True for parallel execution, false for sequential execution
output	Output stream to print status messages

nbSCDOFs()

size_t HArDCore2D::KirchhoffLove::nbSCDOFs ( ) const

inline

Returns the number of statically condensed DOFs (here, the cell moments DOFs)

polykm2Th()

const GlobalDOFSpace HArDCore2D::KirchhoffLove::polykm2Th ( ) const

inline

Returns the space \(\mathbb{P}^{k-2}(\mathcal{T}_h)\).

scMatrix()

const SystemMatrixType & HArDCore2D::KirchhoffLove::scMatrix ( ) const

inline

Returns the static condensation recovery operator.

scVector()

Eigen::VectorXd & HArDCore2D::KirchhoffLove::scVector ( )

inline

Returns the static condensation rhs.

sizeSystem()

size_t HArDCore2D::KirchhoffLove::sizeSystem ( ) const

inline

Returns the size of the statically condensed system.

stabilizationParameter() [1/2]

double & HArDCore2D::KirchhoffLove::stabilizationParameter ( )

inline

Returns the stabilization parameter.

stabilizationParameter() [2/2]

const double & HArDCore2D::KirchhoffLove::stabilizationParameter ( ) const

inline

Returns the stabilization parameter.

systemMatrix() [1/2]

SystemMatrixType & HArDCore2D::KirchhoffLove::systemMatrix ( )

inline

Returns the linear system matrix.

systemMatrix() [2/2]

const SystemMatrixType & HArDCore2D::KirchhoffLove::systemMatrix ( ) const

inline

Returns the linear system matrix.

systemVector() [1/2]

Eigen::VectorXd & HArDCore2D::KirchhoffLove::systemVector ( )

inline

Returns the linear system right-hand side vector.

systemVector() [2/2]

const Eigen::VectorXd & HArDCore2D::KirchhoffLove::systemVector ( ) const

inline

Returns the linear system right-hand side vector.

xDivDiv()

const XDivDiv & HArDCore2D::KirchhoffLove::xDivDiv ( ) const

inline

Returns the space XDivDiv.

Variable Documentation

biquartic_divdiv_hess_u

KirchhoffLove::ForcingTermType HArDCore2D::biquartic_divdiv_hess_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return 24*pow(x(0), 2)*pow(x(0) - 1, 2) + 32*x(0)*x(1)*(x(0) - 1)*(x(1) - 1) + 8*x(0)*x(1)*(x(0) - 1)*(4*x(1) - 2) + 8*x(0)*x(1)*(4*x(0) - 2)*(x(1) - 1) + 8*x(0)*x(1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 8*x(0)*(x(0) - 1)*(x(1) - 1)*(4*x(1) - 2) + 8*x(0)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 24*pow(x(1), 2)*pow(x(1) - 1, 2) + 8*x(1)*(x(0) - 1)*(4*x(0) - 2)*(x(1) - 1) + 8*x(1)*(x(0) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 2*(4*x(0) - 4)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1));
  }

biquartic_dx_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::biquartic_dx_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << 2*pow(x(1), 2)*(12*x(0) - 6)*pow(x(1) - 1, 2), 4*x(0)*x(1)*(x(0) - 1)*(x(1) - 1)*(4*x(1) - 2) + 4*x(0)*x(1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 4*x(1)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1));
    M.row(1) << 4*x(0)*x(1)*(x(0) - 1)*(x(1) - 1)*(4*x(1) - 2) + 4*x(0)*x(1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 4*x(1)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)), 2*pow(x(0), 2)*(2*x(0) - 2)*(pow(x(1), 2) + 4*x(1)*(x(1) - 1) + pow(x(1) - 1, 2)) + 4*x(0)*pow(x(0) - 1, 2)*(pow(x(1), 2) + 4*x(1)*(x(1) - 1) + pow(x(1) - 1, 2));
 
    return M;
  }

biquartic_dy_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::biquartic_dy_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << 2*pow(x(1), 2)*(2*x(1) - 2)*(pow(x(0), 2) + 4*x(0)*(x(0) - 1) + pow(x(0) - 1, 2)) + 4*x(1)*pow(x(1) - 1, 2)*(pow(x(0), 2) + 4*x(0)*(x(0) - 1) + pow(x(0) - 1, 2)), 4*x(0)*x(1)*(x(0) - 1)*(4*x(0) - 2)*(x(1) - 1) + 4*x(0)*x(1)*(x(0) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 4*x(0)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1));
    M.row(1) << 4*x(0)*x(1)*(x(0) - 1)*(4*x(0) - 2)*(x(1) - 1) + 4*x(0)*x(1)*(x(0) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)) + 4*x(0)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)), 2*pow(x(0), 2)*pow(x(0) - 1, 2)*(12*x(1) - 6);
 
    return M;
  }

biquartic_grad_Delta_u

KirchhoffLove::GradientDeflectionType HArDCore2D::biquartic_grad_Delta_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
  
    Eigen::Vector2d G {
      biquartic_dx_hess_u(x)(0,0)+biquartic_dx_hess_u(x)(1,1), 
      biquartic_dy_hess_u(x)(0,0)+biquartic_dy_hess_u(x)(1,1)
      };
    
    return G;
  }

biquartic_grad_u

KirchhoffLove::GradientDeflectionType HArDCore2D::biquartic_grad_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
    return VectorRd(pow(x(0), 2)*pow(x(1), 2)*pow(1 - x(1), 2)*(2*x(0) - 2) + 2*x(0)*pow(x(1), 2)*pow(1 - x(0), 2)*pow(1 - x(1), 2), pow(x(0), 2)*pow(x(1), 2)*pow(1 - x(0), 2)*(2*x(1) - 2) + 2*pow(x(0), 2)*x(1)*pow(1 - x(0), 2)*pow(1 - x(1), 2));
  }

biquartic_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::biquartic_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
      M.row(0) << 2*pow(x(1), 2)*pow(x(1) - 1, 2)*(pow(x(0), 2) + 4*x(0)*(x(0) - 1) + pow(x(0) - 1, 2)), 4*x(0)*x(1)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1));      
      M.row(1) << 4*x(0)*x(1)*(x(0) - 1)*(x(1) - 1)*(x(0)*x(1) + x(0)*(x(1) - 1) + x(1)*(x(0) - 1) + (x(0) - 1)*(x(1) - 1)), 2*pow(x(0), 2)*pow(x(0) - 1, 2)*(pow(x(1), 2) + 4*x(1)*(x(1) - 1) + pow(x(1) - 1, 2));
    return M;
  }

biquartic_hess_u_DE

KirchhoffLove::MomentTensorEdgeDerivativeType HArDCore2D::biquartic_hess_u_DE

static

Initial value:

= [](const VectorRd & x, const Edge & E) -> double {
 
    Eigen::Vector2d tE = E.tangent();
    Eigen::Vector2d nE = E.normal();
 
    Eigen::Matrix2d dtE_hess = biquartic_dx_hess_u(x) * tE.x() + biquartic_dy_hess_u(x) * tE.y();
 
    Eigen::Vector2d div_hess {
      biquartic_dx_hess_u(x)(0,0) + biquartic_dy_hess_u(x)(0,1),
      biquartic_dx_hess_u(x)(1,0) + biquartic_dy_hess_u(x)(1,1),
    };
 
    return (dtE_hess*nE).dot(tE) + div_hess.dot(nE);
  }

biquartic_u

KirchhoffLove::DeflectionType HArDCore2D::biquartic_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return pow(x(0),2)*pow(1.-x(0),2)*pow(x(1),2)*pow(1.-x(1),2);
  }

PI

const double HArDCore2D::PI = boost::math::constants::pi<double>()

static

quartic_divdiv_hess_u

KirchhoffLove::ForcingTermType HArDCore2D::quartic_divdiv_hess_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return 48.;
  }

quartic_dx_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::quartic_dx_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << 24*x(0) - 12, 0;
    M.row(1) << 0., 0.;
 
    return M;
  }

quartic_dy_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::quartic_dy_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << 0., 0.;
    M.row(1) << 0., 24*x(1) - 12;
 
    return M;
  }

quartic_grad_Delta_u

KirchhoffLove::GradientDeflectionType HArDCore2D::quartic_grad_Delta_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
  
    Eigen::Vector2d G {
      quartic_dx_hess_u(x)(0,0)+quartic_dx_hess_u(x)(1,1), 
      quartic_dy_hess_u(x)(0,0)+quartic_dy_hess_u(x)(1,1)
      };
    
    return G;
  }

quartic_grad_u

KirchhoffLove::GradientDeflectionType HArDCore2D::quartic_grad_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
    return VectorRd(pow(x(0), 2)*(2*x(0) - 2) + 2*x(0)*pow(1 - x(0), 2), pow(x(1), 2)*(2*x(1) - 2) + 2*x(1)*pow(1 - x(1), 2));
  }

quartic_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::quartic_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
 
    Eigen::Matrix2d M;
    M <<
      2.*(pow(x(0),2)+4.*x(0)*(x(0)-1.)+pow(1.-x(0),2)), 0.,
      0., 2.*(pow(x(1),2)+4.*x(1)*(x(1)-1.)+pow(1.-x(1),2));
 
    return M;
  }

quartic_hess_u_DE

KirchhoffLove::MomentTensorEdgeDerivativeType HArDCore2D::quartic_hess_u_DE

static

Initial value:

= [](const VectorRd & x, const Edge & E) -> double {
 
    Eigen::Vector2d tE = E.tangent();
    Eigen::Vector2d nE = E.normal();
 
    Eigen::Matrix2d dtE_hess = quartic_dx_hess_u(x) * tE.x() + quartic_dy_hess_u(x) * tE.y();
 
    Eigen::Vector2d div_hess {
      quartic_dx_hess_u(x)(0,0) + quartic_dy_hess_u(x)(0,1),
      quartic_dx_hess_u(x)(1,0) + quartic_dy_hess_u(x)(1,1),
    };
 
    return (dtE_hess*nE).dot(tE) + div_hess.dot(nE);
  }

quartic_u

KirchhoffLove::DeflectionType HArDCore2D::quartic_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return pow(x(0),2)*pow(1.-x(0),2) + pow(x(1),2)*pow(1.-x(1),2);
  }

trigonometric_divdiv_hess_u

KirchhoffLove::ForcingTermType HArDCore2D::trigonometric_divdiv_hess_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return 4.*pow(PI, 4)*sin(PI*x(0))*sin(PI*x(1));
  }

trigonometric_dx_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::trigonometric_dx_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << -cos(PI*x(0))*sin(PI*x(1)), -sin(PI*x(0))*cos(PI*x(1));
    M.row(1) << -sin(PI*x(0))*cos(PI*x(1)), -cos(PI*x(0))*sin(PI*x(1));
 
    return pow(PI, 3) * M;
  }

trigonometric_dy_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::trigonometric_dy_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << -sin(PI*x(0))*cos(PI*x(1)), -cos(PI*x(0))*sin(PI*x(1));
    M.row(1) << -cos(PI*x(0))*sin(PI*x(1)), -sin(PI*x(0))*cos(PI*x(1));
 
    return pow(PI, 3) * M;
  }

trigonometric_grad_Delta_u

KirchhoffLove::GradientDeflectionType HArDCore2D::trigonometric_grad_Delta_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
  
    Eigen::Vector2d G {
      trigonometric_dx_hess_u(x)(0,0)+trigonometric_dx_hess_u(x)(1,1), 
      trigonometric_dy_hess_u(x)(0,0)+trigonometric_dy_hess_u(x)(1,1)
      };
    
    return G;
  }

trigonometric_grad_u

KirchhoffLove::GradientDeflectionType HArDCore2D::trigonometric_grad_u

static

Initial value:

= [](const VectorRd & x) -> VectorRd {
    return PI * VectorRd(sin(PI*x(1))*cos(PI*x(0)), sin(PI*x(0))*cos(PI*x(1)));
  }

trigonometric_hess_u

KirchhoffLove::MomentTensorType HArDCore2D::trigonometric_hess_u

static

Initial value:

= [](const VectorRd & x) -> Eigen::Matrix2d {
    Eigen::Matrix2d M;
    M.row(0) << -sin(PI*x(0))*sin(PI*x(1)), cos(PI*x(0))*cos(PI*x(1));
    M.row(1) << cos(PI*x(0))*cos(PI*x(1)), -sin(PI*x(0))*sin(PI*x(1));
 
    return pow(PI, 2) * M;
  }

trigonometric_hess_u_DE

KirchhoffLove::MomentTensorEdgeDerivativeType HArDCore2D::trigonometric_hess_u_DE

static

Initial value:

= [](const VectorRd & x, const Edge & E) -> double {
 
    Eigen::Vector2d tE = E.tangent();
    Eigen::Vector2d nE = E.normal();
 
    Eigen::Matrix2d dtE_hess = trigonometric_dx_hess_u(x) * tE.x() + trigonometric_dy_hess_u(x) * tE.y();
 
    Eigen::Vector2d div_hess {
      trigonometric_dx_hess_u(x)(0,0) + trigonometric_dy_hess_u(x)(0,1),
      trigonometric_dx_hess_u(x)(1,0) + trigonometric_dy_hess_u(x)(1,1),
    };
 
    return (dtE_hess*nE).dot(tE) + div_hess.dot(nE);
  }

trigonometric_u

KirchhoffLove::DeflectionType HArDCore2D::trigonometric_u

static

Initial value:

= [](const VectorRd & x) -> double {
    return sin(PI*x(0))*sin(PI*x(1));
  }